Here are 100 chapter titles for a book on Network Flows, progressing from beginner to advanced, with a focus on the underlying mathematics:
I. Foundations (1-20)
- Introduction to Network Flows: What and Why?
- Graphs and Networks: Basic Definitions
- Directed and Undirected Graphs: Representing Relationships
- Graph Representations: Adjacency Matrix and Adjacency List
- Paths, Cycles, and Connectivity: Fundamental Concepts
- Network Terminology: Sources, Sinks, and Capacities
- Flow Conservation: The Heart of Network Flows
- Max-Flow Problem: Maximizing Flow Through a Network
- Cuts in Networks: Dividing the Graph
- Max-Flow Min-Cut Theorem: A Fundamental Duality
- The Ford-Fulkerson Algorithm: A Classic Approach
- Residual Graphs: Visualizing Flow Augmentations
- Augmenting Paths: Finding Paths with Available Capacity
- Edmonds-Karp Algorithm: Efficient Max-Flow
- Capacity Scaling: Another Efficient Max-Flow Algorithm
- Minimum Cut Problem: Finding the Smallest Cut
- Relationship between Max-Flow and Min-Cut
- Applications of Network Flows: An Overview
- Real-World Examples: Transportation, Logistics, and More
- Review and Preview: Looking Ahead
II. Intermediate Techniques (21-40)
- Variations on Max-Flow: Multiple Sources and Sinks
- Networks with Vertex Capacities: Constraints on Nodes
- Lower Bounds on Arc Flows: Minimum Flow Requirements
- Circulation Problem: Flows with Demands and Supplies
- Minimum Cost Flow Problem: Minimizing Flow Cost
- Linear Programming Formulation of Network Flows
- Duality in Network Flows: Primal and Dual Problems
- Complementary Slackness: Optimality Conditions
- Shortest Path Algorithms: Dijkstra's and Bellman-Ford
- Minimum Cost Flow Algorithms: Cycle Canceling
- Successive Shortest Path Algorithm: Another Approach
- Network Simplex Method: A Specialized Algorithm
- Integrality Theorem: Flows are Often Integer-Valued
- Unimodularity: Properties of Network Flow Matrices
- Total Unimodularity: Implications for Integer Solutions
- Bipartite Matching: A Network Flow Application
- Assignment Problem: Matching Workers to Tasks
- Transportation Problem: Distributing Goods Efficiently
- Transshipment Problem: Intermediate Nodes Allowed
- Review and Practice: Intermediate Techniques
III. Advanced Topics (41-60)
- Push-Relabel Algorithm: A More Sophisticated Approach
- Generic Push-Relabel: Framework and Analysis
- Relabel-to-Front: A Specific Implementation
- Highest-Label Push-Relabel: Another Variant
- Preflows: Relaxing Flow Conservation
- Gap Heuristic: Improving Push-Relabel Efficiency
- Dynamic Graphs: Flows Changing Over Time
- Maximum Flow in Dynamic Graphs
- Minimum Cut in Dynamic Graphs
- Data Structures for Network Flows: Efficient Implementations
- Fibonacci Heaps: Improving Dijkstra's Algorithm
- Dynamic Trees: Maintaining Connectivity Information
- Scaling Algorithms: Polynomial Time Solutions
- Approximation Algorithms: Dealing with Hard Problems
- Randomized Algorithms: Probabilistic Approaches
- Parallel Algorithms: Solving Network Flows Concurrently
- Distributed Algorithms: Flows in Distributed Networks
- Multicommodity Flows: Flows of Multiple Goods
- Concurrent Flows: Maximizing Flow Rates
- Review and Practice: Advanced Topics
IV. Special Topics and Applications (61-80)
- Network Flow Applications in Communication Networks
- Network Routing: Finding Optimal Paths
- Congestion Control: Managing Network Traffic
- Network Flow Applications in Supply Chain Management
- Logistics and Transportation: Optimizing Distribution
- Scheduling Problems: Resource Allocation
- Project Management: Critical Path Method
- Image Segmentation: Graph Cuts
- Maximum Likelihood Estimation: Network Flow Formulation
- Network Flow Applications in VLSI Design
- Circuit Layout: Minimizing Wire Length
- Network Reliability: Assessing Network Robustness
- Network Security: Protecting Against Attacks
- Social Networks: Analyzing Connections and Influence
- Epidemiology: Modeling Disease Spread
- Bioinformatics: Gene Expression Analysis
- Game Theory: Network Games
- Combinatorial Optimization: Network Flow Techniques
- Linear Programming: Network Flow Connections
- Advanced Applications: A Survey
V. Deeper Dive and Extensions (81-100)
- Matroids and Network Flows: Generalizing Concepts
- Submodular Functions: Connections to Cuts and Flows
- Electrical Flows: Analogy to Electrical Circuits
- Planar Graphs: Duality and Network Flows
- Geometric Network Flows: Flows in Geometric Spaces
- Parametric Network Flows: Flows as a Function of a Parameter
- Sensitivity Analysis: How Changes Affect Optimal Flows
- Robust Network Flows: Dealing with Uncertainties
- Online Network Flows: Decisions Made Over Time
- Dynamic Programming and Network Flows
- Approximation Algorithms for Network Flows: Advanced Topics
- Interior-Point Methods for Network Flows
- Cutting Plane Methods for Network Flows
- Lagrangian Relaxation for Network Flows
- Network Flow Software and Libraries
- Computational Complexity of Network Flow Algorithms
- History of Network Flows: A Detailed Account
- Open Problems and Future Directions in Network Flows
- Research Topics in Network Flows: A Guide for Exploration
- Advanced Topics in Combinatorial Optimization: Network Flows and Beyond